Normal forms of the degenerate autonomous differential equations with the maximal Jordan chain and simple Applications Текст научной статьи по специальности «Математика»

MSC 34G20

DOI: 10.14529/mmp170301

NORMAL FORMS OF THE DEGENERATE AUTONOMOUS DIFFERENTIAL EQUATIONS WITH THE MAXIMAL JORDAN CHAIN AND SIMPLE APPLICATIONS

L.R. Kim-Tyan1, B.V. Loginov2, Yu.B. Rousak3

1 National University of Science and Technology "MISIS"(MISIS), Moscow, Russian

Federation,

2

3

Canberra, Australia

E-mail: kim-tyan@yandex.ru, bvllbv@yandex.ru, irousak@gmail.com

Degenerate differential equations, as part of the differential-algebraic equations, the last few decades cause increasing interest among researchers, both because of the attractiveness of the considered theoretical questions, and by virtue of their applications. Currently, advanced methods developed in this area are used for system modelling and analysis of electrical and electronic circuits, chemical reaction simulations, optimization theory and automatic control, and many other areas. In this paper, the theory of normal forms of differential equations, originated in the works of Poincare and recently developed in the works of Arnold and his school, adapted to the simplest case of a degenerate differential equations. For this purpose we are using technique of Jordan chains, which was widely used in various problems of bifurcation theory. We study the normal forms of degenerate differential equations in the case of the existence of the maximal Jordan chain. Two and three dimensional spaces are studied in detail. Normal forms are the simplest representatives of the degenerate differential equations, which are equivalent to more complex ones. Therefore, normal forms should be considered as a model type of degenerate differential equations.

Keywords: degenerate differential equations; normal forms; Jordan chains.

Introduction

The aim is to develop a technique of construction of normal forms for systems of differential equations with a degenerate operator at the derivative:

Unless otherwise stated, it is assumed that A, F : E\ — E2, dimEi = n, dimE2 = n, (n=2,3 and 4) where A is a degenerate operator such as dimker A = 1 and ker A = p, ker A* = Let fonction F be sufficiently smooth and B (y) be the linear part of the function F calculated at the point x = 0

These results suggest the presence of Jordan chain consisting of n elements for the operator-function A — £B0(B0 = B (0)) and they are based on the symbiosis of ideas and

Ax' = F (x,ß), F (0,ß) = 0.

(1)

F(x, ß) = B(ß)x + R(x, ß), \\R(x, ß) II = o(||x||).

approaches set forth in [1] and [2] for systems of ordinary differential equations without degeneration. Fundamentals of the theory of normal forms can be found in [3] and [4]. The work consists of two parts: the first part is algebraic in nature. Here the language of A-deformation of the operator B0 = B(0) [2] and the apparatus of generalized Jordan chains (GJC) [1] are used to investigate the perturbations which do not change the Jordan structure of the operator-function. The second part of the work is presented in [5] and devoted to normal forms and bifurcations in the case of maximal Jordan chain, which consists of 2, 3 or 4 elements. In constructing the normal forms for non-autonomous differential equations has been used the apparatus of the differential Jordan chains (DJC). Technically complicated case n = 4 for non-autonomous differential equations which use the DJC and vector spaces over rings of polynomials (modules) is omitted, and published in the local press [6], and in [7]. It turned out that this type of the system describes aero elastic model for transonic circulating of gas flow around plates and envelopes [8].

1. The Normal Form of the Non-Autonomous Degenerate Differential Equations, Depending on Parameters. Perturbations Which Keep the Jordan Structure of Equation

Definition 1. The operator-function B (f) is called an A-deformation of the operator-Bo = B(0) if for small f the operator-function A — eB (f) has the same Jordan structure as the operator-function A — eB0.

Further in this article we consider that the operator-function A — eB0 has the maximal Jordan chain: p(1 = p, p(2\ ..., p(p\ so p = n.

Lemma 1. If the operator-function A — eB0 has the maximal Jordan chain, then B0 is an invertible operator.

Proof The elements {p^^of the Jordan chain of the operator-function A — eB0 are defined by relations

Ap(k+1) = Bop(k), (Bop(k),^(1) ) = 0, k = l,p — 1; (Bop(p),4> ) = 0. (2)

Because p = n elements of the chain form a basis of E^ If the operator B0 is not invertible then exists an element u of the space E1 such as B0u = 0. Let u = £ip(1)+... +$a-ip(n-1) + £np(n) then iiBop(l)+ ... +in-iBop(n-1) + £nBop(n) = 0. From the condition (^B0p(k'), = Skn follows that £n = 0. Then $t1B0p(1'1 + ...

+£n-1B0p(n-1) = A(£1 p(2)+... +£n-1p(n)) = 0 and therefore £1p{2)+.. ■ +£n-1p(n) = Xp{1) (because ker A = p). Since the elements p(l\ ..., p(n form the basis of the space, the ^ = ^^ _ ., £n-1 = 0 and u = 0.

□

Corollary 1. If the B(f) is A-deformation of B0, then the equation (1) can be written as

A(f)x' = x + R1 (x,f), (3)

where the matrix of the operator A(f) = B-1(f)A has the fo rm [A(f)] = [C (f)]J [C (f)]-1. Here, the matrix [C(f)] is formed by the columns of [p(l\ p(2)(f), ...,p(n(f)], and J is Jordan block.

Proof. Let us apply the matrix [C(y)] l[B 1(y)A][C(y)] to the vector ei = (0,..., 1,..., 0), [C(y)]-1[B-1(y)A][C(y)]ez = [C(y)]-1[B-1(y)A]'(i)(y) = [C(y)]-1'i-1)(y) = e-1, if i > ^en i = 1, then [C(y)]-1[B-1(y)A][C(y)]e1 = 0. Therefore [C(y)]-1[B-1(y)A][C(y)] = J.

□

Lemma 2. Let matrix [C(y)] be a deformation of E (identical matrix), where the first column has the form [C(y)]e1 = a(y)', a(y) = 0 a(0) = 1. Then [C(y)] defines the operator-function B(y) - an A-deformation of the operator B0 - up to the zero vector-deformation X (y), X (0) = 0.

Proof. Indeed, let the matrix [C(y)] be such that [C(0)] = Then for

sufficiently small y matrix [C(y)] is not degenerate, and vectors '^(y) = [C(y)]ei form a basis of the space E^ We define a mapping B(y) % formulas (using multiplier 1/a(y))\ B(y)'(i)(y) = A'i+1)(y) when i < n, and B(y)'(n)(y) = B(0)'p) + X(y). Here X(y) be an arbitrary deformation of vector 0. The operator-function A — eB(y) has a maximal Jordan chain (y), ...,'(n\y). Further, instead of operators we will consider their

matrix in some basis {e1 ,e2,... ,en}.

□

Definition 2. The operator-function B(y) is called A-versal deformation of the operator-Bo = B (0), if any A-deformation P(v) of the opera tor B0, can be obtained from B (y) by replacing the parameter, i.e., there exists differentiable mapping y = k(v), k(0) = 0, and the deformation [5(v)\ of the identical matrix E, such that

IP-1 (v)A] = [S (v)][B-1(k(v))A][S (v)]-1. (4)

Remark 1. It follows from (4) that the tp(1 is an eigenvector of the matrix [S(v)]. Indeed, A'/1 =0 ^ A[S(v)]-1'(1) = 0 ^ [S(v)]-1'(1) = X'(1).

Lemma 3. Let W be the subgroup of GL (n, R) of reversible matrix stretching vector '(WS E W : S' = XX = 0) mid Ev sub spa ce gl (n, R) of all matrices for which ' is an eigenvector. Through Yw (B) denote the orbit of matrix B under the action of W : yw(B) = {SBS-1,S E W}. Then

tB (yW (B )) = Im(LB\EV), (5)

where TB(yw(B)) is the tangent space to the orbit jw(B) at the point B and LB\E^ be the mapping defined by the formula [v,B]. Here v E Ev, LB = [v,B] is the commutator of vB

Proof. If matrix S is close to E (identical) then S belongs to W if and only if S = E + u, where u E Ev (u is small enough). Indeed if S E W then (S — E)' = (X — 1)' and u E Ev ,conversely if u E E^ then S' = (1 + y)' , 1 + y = 0 because S is invertible. Now

(E + eu)B(E + en)-1 = (E + eu)B (E — eu + ...) = B + e[u, B ] + which proves (5).

□

Remark 2. codim(YW(B)) = dim(ZB fi Ev) + codim(E^). Here ZB = kerLB is the

B

Indeed, codim(YW(B)) = dim gl (n, R) — dim(YW(B)) = dim gl (n, R) — — dimIm(LB\EV), but dimIm(LB\E^) = dim Ev — dim ker(LB\EV) = dim Ev—

— dim (ZB H Ev). Consequently, codim (yw(B)) + dim (Zb H Ev) = dim (Zb H Ev) + codim (Ev).

dim gl (n, R) — dim Ev+

Corollary 2. If V is a manifold which belongs to Ev, contains zero matrix O and intersects transversally ZBHE^ at the point 0 then LB(T0V) = TB(jw(B)). Here T0V is the tangent space to the manifold V at the po int 0.

Proof. By the transversality, for each Wu E Ev 3u1 E T0V and u2 E T0(ZB H Ev) such that u = ui + u2 ^ LB(u) = LB(u1 + u2) = LB(ui) and because of u2 E ZBl Im (LB\T0V) = Im (Lb\Ev) = Tb(yw(B)).

□

Remark 3. If B = J, then Ev consists of matrices in which the first column has the form (a, 0,..., 0), and thus, dim Ev = n2 — n + 1. On the other hand, it is known [2], that Zj is Tj (yw (J)) span of mat rices E, J, J2, ... , Jn-1, each of which belongs Ev so dim TJ(jw(J)) = n2 — 2n + 1 So V can be selected as subspace matrices

( 0

0

0

Vl1

0 \

Vln-1

(6)

\ 0 Vn-11 . . . Vn-1n-1 J

Then the space TJ(jw(J)) ^^^^^^te of matrices of the form vJ — Jv.

Indeed, by Corollary 2, Im (LJ\T0V) = Im (LJ\E^) = TJ(jw(J)), where V is a submanifold of Ev contains zero matrix O and transversal to ZJ H Ev at zero.

Lemma 4. Let B : A ^ gl(n,R) E C1, where A is neighbourhood of zero in №.k, B = B (A) , B(0) = B0. We assume that B (X) transversal to jw (B0) at the po int A = 0 and k = codim (yw(B0)) and V is a manifold which belongs to Ev, contains zero matrix O and intersects transversally ZB H Ev at the po int 0 and such as dim V = dim yw (B0). Then the map $(v, A) : V x A ^ gl (n, R), defined by the formula $(v, A) = (E + v)B(A)(E + v)-1, sets the local diffeomorphism of a neighborhood of the point (0, 0) m V x A.

Proof. Let us calculate the derivative of $(v, A) with respect to v at the point (0, 0). It follows from the formula (E + ev)B(A)(E + ev)-1 = (E + ev)B(A)(E — ev + ... ) = B(A) + e[v,B(A)] + ... that derivative of $(v,A) at ^te point (0, 0) is [-,B]. According to Corollary 2 it maps the tangent space to the manifold V to the tangent space to Yw(B0). The derivative of $(v, A) with respect to A at the point (0, 0) equals B'(0) - an arbitrary matrix from the tangent space TB(0) (tangent space to B(A ) at A = 0). Due to the manifold V is transversal to ZB H Ev at the po int 0 the derivative of $(v, A) at the point (0, 0) maps the tangent space to V x A to the whole space gl (n, R). But, according to suppositions k = codim (yw(B0)) and dim V = dim (yw(B0)), so dim V + dim A = dim gl (n, R) and therefore $(v, A) is the local diffeomorphism.

□

Lemma 5. If B = B-1A, and U is subspace of E^ such that Ev = (ZB H Ev) © U, then the mapping : U ^ gl (n, R), defined by the formula = (E + u)B(E + u)-1

U

of B in Yw (B).

Proof. As in lemma 4 derivative of the mapping ^(u) is equal to [•,B] and, by virtue of (5), it maps U to TB(jw(B)). By construction, it has no zeros in U and is therefore an isomorphism. Then ^(u) is a local diffeomorphism.

□

Theorem 1. B(y) is A-deformation of B0 = B(0) ), if and only if [B-1(y)A] belongs to the submanifold jw (B-1A).

Proof. Necessity. By the Corollary 1, the matrix [B-1(y)A] is represented as [C(y)]J[C(y)]-1 and respectively [B-1(0)A] = [C(0)]J[C(0)]~\ i.e. [B-1(y)A] = [C(y)][C(0)]~1[B-1A][C(0)][C(y)]-\ In this case [C(y)][C(0)]-1^ = \(y)<p, X(y) = 0, and consequently, [B-1(y)A] belongs to the submanifold yW(B0 lA).

Sufficiency. If [B-1(y)A] belongs to the submanifold

Yw(B-1A), then for any y [B-1(y)A] = [C(y)][B- A][C(y)]-1 = [C (y)][C (0)]-1 • •[B-1(0)A][C(0)][C(y)]-1 = [S(y)]J[S(y)]-^^ ^^^emma 2, B(y) is A-deformation of B0. Continuity (or differentiability if B(y) is differentiate) of C(y) follows from Lemma 5. In this case C(y) has the form E + u(y), where u(y) belongs to the subspace U

□

Corollary 3. Let B(y) be an A-deformation of B0 = B(0). Then B(y) is a versal A

Proof. Indeed, let A(v) be other A-deformation of B0 = A(0) then [A~l(v)A] = [C(v)][B-1 A][C(v)]-1 ■ Introducing the function k(v) = y, we obtain [A-1(v)A] = [CC(v)][B(k(v))-1A][C(v)]-1, where [B(n(v))-1A] = B-1A.

□

Corollary 4. Let B-1 A = J, and y = (y1y y2, ■ ■ ■, yk) be a vector parameter, in order-to B(y) be a versal A-deformation of B0, it is necessary to all dB-1(0)A/dyi to have the following form. Let U be arbitrary matrix (6) of size (n — l)2, and JJi be n-column (u1i, ■ ■ ■ ,un-1i, 0), and Ui be n-colu,mn (0,u1i, ■ ■ ■ ,un-1i). Then, according to Remark 3, matrix belonging to Tj(jw(J)) must have the form [0, —U1, U1 — U2, ■ ■■, Un-2 — Un-1].

Remark 4. This condition is not sufficient.

Indeed, the matrix U = f 1 1 ] generates a third-order matrix I 0 10 | which

\ 1 1 J \0 0 1

could not be transformed to the single Jordan cell, because it has two different eigenvalues 01

Theorem 2. Let the B-1 A = J and Eis be the matrix in which 1 is in place (i,s) and zeros are at the other places, and s > i + 1. Then, for small £ the matrix J + eEis belongs to Yw (J) an d Eis belongs to the tangent s ubspace Tj (jw (J ))•

Proof. Indeed, in the standard basis matrix J + £Eis acts от it as follows: 0 ^ e1 ^ e2 ^ ••• ^ es-i; es-i + £ei ^ es ...; en_i ^ en, in the basis ф1 = ei;...; ps-i = es-i; = es - £ei+i;...; ф2s-i-2 = e2s-i-2 - £e.s-i; <^2s-i-i = e2s-i-i - £e.s + £ e+i;... it turns into the Jordan block.

_a

Вестник ЮУрГУ. Серия «Математическое моделирование Q

и программирование» (Вестник ЮУрГУ ММП). 2017. Т. 10, № 3. С. 5-15

0 1 -1

Remark 5. Any matrix, which has form

belongs to Yw (J)•

( 0

0

0

V 0

i 0

0 0

*

i

0 0

i 0

Indeed, it has one eigenvalue equal to zero and one eigenvector e1, therefore, it could be reduced to the Jordan block.

Definition 3. Let A(v) be A-deformation of B0 = B(0). We call A(v), induced from, B(y) (B(y) is A-deformation of B0 = B(0))), with the help of the map y = k(v), if [A-1 (v )A] = [B-1(k(v ))A],

Definition 4. Let B(y) be A-deformation of B0 = B(0). We call B(y) generating if any other A-deformation of B0 (sa y A(v)) induced from B (y) by means of some map y = k(v ).

Theorem 3. If the dimension of the tangent space to the orbit of B-1(y)A at the point y = 0 is equal to the dimension of th e sub space U (introduced in Lemma 5) then the deformation B-1A is generating.

Proof. It follows from the Lemma 5 that the dimension of the tangent space to the

B-1(0)A y = 0

orbit yw(B-1A). Therefore the mapping B-1(y)A defines a local diffeomorphism from the neighbourhood of the point y = 0 to the neighbour hood B-1A in orbit Yw (B-1A).Let r be the inverse map such that for small y, y = r(B-1(y)A). Then, if A(v) is an arbitrary A-deformation of B0 = B(0), we can construct the mapping y = k(v) as y = r(A-1(v)A). Receives [A-1 (v)A] = [B-1(k(v))A].

1 1 □

Remark 6. If the perturbation B 1(y)A of B0 A is known, then it is possible to build the corresponding perturbation B(y) of the operator B0 as follows.

Let B-1(y)A = B-1A + S(y), or A = B(y)(B-1A + S(y)). Since the perturbation B-1A + S(y) corresponds to some A-deformation of B0, there is the maximal Jordan chain p(1),p(2)(y),...,p(n)(y), for which (B-1A+S (y))p(1) = 0 p(l)(y) = (B-1A+S (y))p(i+1)(y) (1 < i < n). Therefore, if Y(y) is the functional, determined by the conditions (p(1'),Y(y)) = 1 i^(k)(y),Y(y)) = 0 k = 2,..., n, then the operator D(y) = B--1A + S(y) + (^,Y(y))^(n)(y) is reversible [1]. Operator D(y) is called the Schmidt's regularization for operator B-1(y)± Then A = B(y)[D(y) — (•,Y(y))p(n)(y)] = B(y)D(y) — (•,Y(y))B(y)p(n)(y), and by Lemma 2 B(y) = AD-1(y) + (D-1(y)^,Y(y))(Bop(n)(0) + X (y))-

2. Normal Forms and Bifurcations in the Case of the Maximal Jordan Chain of Small Length

A. The Jordan Chain of Length Two. Since \\R(x,y)\\ = o(||x||), in the case of the

2

differential equation in the basis of (y)

x'2 = x1 + f (x1,x2,y), 0 = x2 + g(x1 ,x2,y), using reduction of Lyapunov - Schmidt [1].

The functions f (x1,x2,y) and g(x1, x2, y) satisfy conditions: f (0, 0,y) = 0, g(0, 0,y) = 0 h Df (0, 0,y) = 0, Dg(0, 0,y) = 0. Using the change of variables y1 = x1 + f (x1,x2,y), y2 = x2 this system could be simplified y'2 = y^ y2 = h(y1, y2, y). If the function h(y1,y2,y)

y1 (0, 0)

y1 = 0 y2 = 0

implicit function, y2 = G(y1,y), G(0,y) = 0 G'(0,y) = 0 and therefore the system takes the form

y2 = yu y2 = y2ih(y1,y), \h(y1,y)\ = o(\y1\ + |y|). (7)

In the simplest case, h(y1, y) = y1 +y we obtain the normal form of "loss of uniqueness" bifurcation y'2 = y1,y2 = y2(y1 + y)- When y = 0 y1(t) = 0 y2(t) = 0 is the only solution

y=0

equation (3y2 + 2yy1)y'1 = y1 ot if y1 = 0, to (3y1 + 2y)y'1 = 1. Substitution z = 3y1 + 2y, z' = 3y1 allows to simplify the last equation to the form zz' = 3, which has a smooth solution in a neighborhood of t = 0 with initial value z(0) = 2y. Thus, the uniqueness is violated.

y=0 (0, 0)

dh(0, 0)/dy = 0 it is sufficient, that h(0, 0) = 0.

h(0, 0) = 0 (0, 0) h(y1 ,y) = A+y1h1(y1,y)+ yh2(y1,y), A = 0. Differentiating the equation y2 = y^h1(y1,y) t y2 y1

y1 = 0 y

y[ = [2A + 3y1 h1(y1,y) + y2h1(y1,y) + 2yh2(y1,y) + yy1h/2(y1,y)]-1.

y = 0 y

starting from the point (0,0).

Sufficiency. Let the conditions h(0,0) = 0 dh(0,0)/dy = 0. Then h(y1,y) = y1 h1(y1,y) + yh2(y1, y), where h2(0, 0) = hence y2 = yfh1 (y1,y) + yy^h2(y1,y), and y2 = y^h(y1, 0) = yfh(y1) for y = 0. Therefore, the first equation of the system (7) takes the form: [3y2h(y1) + y\h'(y1 )]y1 = y1. This equation has only one solution with initial value y1 (0) = 0, namely y1(t) = ^f y = ^^en y2 = A(y)y^L + yfh3(y1,y), A(y) = 0,

y1 y y2

for y1 = 0 gives [2A(y)y1 + y^h4(y1, y)]y[ = y1 ^ y[ = [2A(y)y1 + y\hA(y1,y)]-1. This

y (0) = 0 y = 0

bifurcation point.

□

B. The Jordan Chain of Length Three. In the case of the Jordan chain of length three, just like in the case A, the degenerate differential equation in the basis of tp( 1 ), tp(2\y), p(3\y) could be transformed to the system:

y2 = y 1, y3 = y2 + y2f2(y 1,y2,y), y3 = f3(y 1,y2,y). (8)

Function f3 satisfies the following conditions f3(0, 0,y) = ^d Df3(0, 0,y) = 0.

Consider the particular case of the system (8)

v2 = Vi, v'S = V2, y3 = mÍ + y2- (9)

Let show that f = 0 is a bifurcation point "changing of the domain of solution". First consider the system (9) in a punctured neighbourhood of f = 0, (f = 0). This system takes the form:

v'2 = Vi, 2fViVi + 2vÍvI = V2, (10)

which is equivalent to the system (when v1 = 0)

V2 = V2, Wi = V2 - 2V2Vi, (11)

because [9] the solutions of both of these systems have the same orbits (when yi = 0 and y2 = 0 ) as the solutions of the equation dy^ = y2-2¿y2yi • The system (11) by the Existence and Uniqueness theorem has solution started from each point (y0,y2), in some neighbourhood of (0, 0). To clarify the behavior of solutions in a neighbourhood of y = 0, ff = 0 we repeat the process of blowing-up of singularity by transition to the new variables (y i,u) where V2 = uy ^ The n y2 = u' • yj + u • 2y 1 (l/2f)(y2 - 2y2y 1) = u' • yj + u • y 1 (l/f)(u • V2 - 2u • v\).

By substituting this into (11) we obtain:

u' =1 - u • (l/f)(u • Vi - 2u • vl), 2fy[ = u • v\ - 2u • y\. (12)

(0, 0) (0, u)

it is necessary to investigate all the singular points of the system (12) on this line. However, the vector field defined by the right of (12), has no singular points on this line, because when yi = 0 it is equal to (l, 0).To study the solutions of the original system go back to the variables yy2 = u • y^. When f = 0 system (9) takes the form y'2 = yi , y'3 = y2, Vs = V2 ^ V2 = Vi) 2yiy2 = y2. If y2 = 0 these systems have the solution yi = 1/2, V2 = v2 + t/2, Vs = (V0 + t/2)2.

Thus, in the first case (f = 0), the manifold of the initial values of the solutions were two-dimensional, and in the second case - one-dimensional.

Remark 7. If for degenerate differential equation, with the maximal Jordan chain consisting from three elements, the non-linear part does not depend on the variable y then it has no solutions in some deleted neighbourhood of the point y = 0.

Indeed, the system (8) then takes the form V2 = yi , y's = y2, ys = fs(y2,f) ^ y'2 = y fS (y2,f)Vl = V2 and since Df3(0,f) = 0, the second equation of the last system could be reduced by y2 when y2 = 0 which gives f4(y2, f)yi = l. It is obvious that this equation could not have small solutions, so y = 0 is the only small solution of the system.

Remark 8. The last result does not take place for the degenerate equation with the maximal Jordan chain consisting of four elements. This is evident from example:

v2 = Vi, y's = V2, v4 = V3, V4 = V2 • Vs ^ v2 = Vi, vS = V2, Vi • Vs + V2 = Vs. By substituting the second equation, we get: y'2 = yi, 2yiy2(l - yi)- i + y22(l - yi)~2y'i = y2-

If y2 = 0, the system takes form y'2 = y^ y2y'1 = (1 — y1)2 — 2y1(1 — y1). The system has a solution, starting at any point (y2,y2), y20 = 0.

Lemma 6. (A sufficient condition for the occurrence of bifurcation "of the changing of the

y=0

domain of solutions" for the system (8) in the neighbourhood of y = 0 y = 0, it is sufficient that the function f3(y1 ,y2, y) has the form f3(y1,y2,y) = h1 (y2,y) + yh2(y1,y2, y), where dh1(0,y)/dy2 = 0 Dh2(0,0,y) = 0 and D(dh2/dy1)(0,0,y) = 0.

Indeed, if the y = 0 then the system (8) takes the form y'2 = y^ y'3 = y2+y2f2(y1,y2, 0), y3 = h1(y2, 0). After substituting the third equation by the second: y1dh1(y2, y)/dy2 = y2 + y2f2(y1 ,y2, 0), we can use the implicit function theorem in order to have y2 = F(y1). Thus, if y = 0 solution of (8) lies on the curve (y1, F(y1)), then the second equation of the system (8) can be written as (dh1/dy2)y1+y(dh2/dy2)y1+y(dh2/dy1)yl1 = y2+y21f2(y1,y2, y)-The system y(dh2/dy1)y1 = y2 + y\f2(y1,y2, y) — (dh/dy2)y1 — y(dh2/dy2)y1, y2 = y1 when y = 0 has a solution in a neighbourhood of (0, 0) starting at any point (y2,y2), to which (dh2/dy1)(y2,y2,y) = 0. Since (dh2/dy1)(0,0,y) = 0 and D(dh2/dy1)(0,0,y) = 0, then by the implicit function theorem, the set of solutions of the equation (dh2/dy1)(y^,y20, y) = 0 is one-dimensional, so there is a bifurcation "of the changing of the domain of solution".

Acknowledgements. This work was financially supported by the Ministry of Education and Science of Russia within the framework of stake tasks 2014/232. The study was conducted with financial support of RFBR, research project No 15-01-08599, 15-4-1-02455p_povolgie_ a.

References

1. Vainberg M.M., Trenogin V.A. Theory of Branching of Solutions of Non-Linear Equations. Leyden, Nordhoof International Publishing, 1974.

2. Arnold V.I. Geometricheskie metody v teoriy obyknovennykh differentsialnykh uravneniy [Geometrical Methods in the Theory of Ordinary Differential Equations]. Moscow, Moscow Center for Continuous Mathematical Education, 1999.

3. Shui-Nee Chow, Chengzhi Li, Duo Wang. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, 1994.

4. Jooss G., Adelmeyer M. Topics in Bifurcation Theory and Applications. Singapore, New Jersey, London, Hong Kong, World Scientific, 1992.

5. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Normal Forms of the Degenerate Differential Autonomous and Non-Autonomous Equations with the Maximal Jordan Chain of Length Two and Three. The Bulletin of Irkutsk State University. Series: Mathematics, 2015, vol. 12, pp. 58-71. (in Russian)

6. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Normal Forms for the Degenerate Non-Autonomous Differential Equations in the Spaces Rn, n = 2, 3, 4. Sbornik nauchnykh trudov "Prikladnaya matematika i mekhanika", Ulyanovsk, 2014, no. 10, pp. 142-160. (in Russian)

7. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Differential Equations with Degenerated Variable Operator at the Derivative. Current Trends in Analysis and Its Applications. Proceedings of the 9th ISAAC Congress, Krakow 2013, 2015, pp. 101-108. DOI: 10.1007/9783-319-12577-0 14

8. Marszalek W. Fold Points and Singularity Induced Bifurcation in Inviscid Transonic Flow. Physics Letters A, 2012, vol. 376, issues 28-29, pp. 2032-2037. D01:10.1016/j.physleta.2012.05.003

9. Stepanov V.V. Kurs differentsialnykh uravneniy [The Course of the Differential Equations]. Moscow, Gosudarstvennoe izdatel'stvo tekhniko-teoreticheskoy literatury, 1950.

Received April 8, 2015

УДК 517.9 Б01: 10.14529/ттр170301

НОРМАЛЬНЫЕ ФОРМЫ ВЫРОЖДЕННЫХ АВТОНОМНЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С МАКСИМАЛЬНОЙ ЖОРДАНОВОЙ ЦЕПОЧКОЙ И ПРОСТЕЙШИЕ ПРИЛОЖЕНИЯ

Л.Р. Ким-Тян1, Б.В. Логинов2, Ю.Б. Русак3

1 Национальный исследовательский технологический университет МИСиС,

г. Москва

2

3

Вырожденные дифференциальные уравнения, как часть алгебро-дифференциальных уравнений, последние десятилетия вызывают все больший интерес среди исследователей, как в силу привлекательности рассматриваемых теоретических вопросов, так и в силу их приложений. В настоящее время развитые в данной области методы используются для системного моделирования и анализа электрических и электронных цепей, моделирования химических реакций, теории оптимизации и автоматического регулирования, а также во многих других областях. В настоящей работе теория нормальных форм дифференциальных уравнений, берущая свое начало в работах А. Пуанкаре, а последнее время развиваемая в работах В.И. Арнольда и его учеников, адаптируется к простейшим случаям вырожденных дифференциальных уравнений. Для этого существенно используется техника жордановых цепочек, давно и широко используемая в различных задачах теории бифуркации. Изучаются нормальные формы вырожденных дифференциальных уравнений в случае существования максимальной жордановой цепочки. Подробно изучаются случаи размерностей 2 и 3. Нормальные формы являются единственно возможными представителями вырожденных дифференциальных уравнений, сводящихся к своей нормальной форме. Поэтому нормальные формы следует считать модельными.

Ключевые слова: вырожденные дифференциальные уравнения; нормальные формы,; жордановы цепочки.

Литература

1. Вайнберг, М.М. Теория ветвления решений нелинейных уравнений / М.М. Вайнберг, В.А. Треногин. - М.: Наука, 1969.

2. Арнольд, В.И. Геометрические методы в теории обыкновенных дифференциальных уравнений / В.И. Арнольд. - М.: Московский центр непрерывного математического образования, 1999.

3. Ван, Д. Нормальные формы и бифуркации векторных полей на плоскости / Д. Ван, Ч. Ли, Ш.-Н. Чоу. - М.: Московский центр непрерывного математического образования, 2005.

4. Jooss, G. Topics in Bifurcation Theory and Applications / G. Jooss, M. Adelmeyer. -Singapore; New Jersey; London; Hong Kong: World Scientific, 1992.

5. Логинов, Б.В. Нормальные формы вырожденных автономных и неавтономных дифференциальных уравнений с максимальной жордановой цепочкой длины два и три / Б.В. Логинов, Ю.Б. Русак, Л.Р. Ким-Тян // Известия Иркутского государственного университета. Серия: Математика. - 2015. - Т. 12. - С. 58-71.

6. Логинов, Б.В. Нормальные формы для вырожденных неавтономных дифференциальных уравнений в пространствах Rn, п=2,3,4 / Б.В. Логинов, Ю.Б. Русак, Л.Р. Ким-Тян // Сборник научных трудов «Прикладная математика и механика:». - Ульяновск: УлГТУ, 2014. - № 10. - С. 142-160.

7. Loginov, B.V. Differential Equations with Degenerated Variable Operator at the Derivative / B.V. Loginov, Yu.B. Rousak, L.R. Kim-Tyan // Current Trends in Analysis and Its Applications. Proceedings of the 9th ISAAC Congress. - 2015. - P. 101-108.

8. Marszalek, WT. Fold Points and Singularity Induced Bifurcation in Inviscid Transonic Flow / W. Marszalek // Physics Letters A. - 2012. - V. 376, issues 28-29. - P. 2032-2037.

9. Степанов, В.В. Курс дифференциальных уравнений / В.В. Степанов. - М.: Рос. изд-во технико-теоретической литературы, 1950.

Луиза Ревмировна Ким-Тян, К&НДИДсХТ физико-математических наук, кафедра «Математика», Национальный исследовательский технологический университет МИСиС(г. Москва, Российская Федерация), kim-tyan@yandex.ru.

Борис Владимирович Логинов, доктор физико-математических наук, профессор, кафедра «Высшая математика», Ульяновский государственный технический университет (г. Ульяновск, Российская Федерация), bvllbv@yandex.ru.

Юрий Борисович Русак, К&НДИДсХТ физико-математических наук, доцент, Департамент социального сервиса (г. Канберра, Австралия), irousak@gmail.com.

Работа выполнена в рамках государственного задания № 2014/232 Минобрнауки России и при поддержке грантов РФФИ 15-01-08599, 15-41-02455р.

Поступила в редакцию 8 а,прем,я 2015 г.